Gaussian Wavepacket Time Evolution
Teach me
i
√C ψ(x,t) Components
Re[√Cψ]
Im[√Cψ]
P(x,t)

Guided Inquiry

1 · The free wavepacket

Here’s our particle. A free particle — no forces anywhere — is built here as a wavepacket: plane waves with momenta spread σ around a mean ℏk₀, added with Gaussian weights A(k). The upper plot draws the real and imaginary parts of ψ(x,t), and the lower plot shows the probability density P(x,t) — where the particle can be found. At t = 0, the packet rests as a single hump at x = 0.

2 · A free particle’s width

Let’s start with something familiar. A classical free object — a coasting ball — keeps its sharp position forever, because nothing pushes or pulls it. This quantum packet is just as free. So when time runs, what do you think its width will do?

3 · Why it spreads

So the hump widened and its peak fell. But why does a free packet spread at all? What’s your best explanation?

4 · Give it momentum

Now let’s add some momentum. The packet is rebuilt at t = 0 with mean momentum k₀ = 1. When time runs, what do you expect the probability hump to do?

5 · A narrower packet

Back to rest for this one. The momentum spread is doubled to σ = 2, so the initial hump in x starts twice as narrow. Compared with the σ = 1 packet, how do you think it spreads?

6 · A heavier particle

One last experiment. Keep σ = 2, but make the particle five times heavier: m = 5. Its spread of momenta is unchanged. How do you think the packet spreads now?

Listen
Parameters
Units: ℏ = 1
k₀ 0.00
σ 1.00
m (mₑ) 1.0
Time Speed 0.2
Display
Wavefunction Components
Re[√Cψ]
Im[√Cψ]
About the Simulation
Physics: This simulation follows the Lecture 6 construction: a free-particle wavefunction is built from A(k), evolved with the lecture's α(t), and displayed as Re[\sqrt{C}\psi], Im[\sqrt{C}\psi], and P(x,t).

Visualization: The upper plot shows the real and imaginary parts of \sqrt{C}\psi(x,t). The lower plot shows the lecture probability density P(x,t) = C|\psi|^2. For each slider choice, the vertical axes are fixed from the initial state rather than changing during the time evolution.

Controls: Adjust k₀, σ, and m. Here σ is the lecture's momentum-space width, so smaller σ produces a broader packet in position space.

Note: Here σ is the width in k-space, not the usual position-space Gaussian width. If the packet spreads beyond the visible x-window, the displayed norm drops below 1.
Equations